Theorem rspcdva 2707

Description: Restricted specialization, using implicit substitution. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Hypotheses

Ref	Expression
rspcdva.1	⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
rspcdva.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
rspcdva.3	⊢ (𝜑 → 𝐶 ∈ 𝐴)

Assertion

Ref	Expression
rspcdva	⊢ (𝜑 → 𝜒)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜒,𝑥   𝜑,𝑥

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem rspcdva

Step	Hyp	Ref	Expression
1		rspcdva.2	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
2		rspcdva.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		rspcdva.1	. . . 4 ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜒))
4	3	adantl 271	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐶) → (𝜓 ↔ 𝜒))
5	2, 4	rspcdv 2704	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → 𝜒))
6	1, 5	mpd 13	1 ⊢ (𝜑 → 𝜒)

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603

This theorem is referenced by:  uzsinds  9428  iseqz  9469  bezoutlemex  10390  bezoutlemzz  10391  bezoutlemmo  10395  bezoutlemle  10397  bezoutlemsup  10398